One of the usual assumptions for the GLM procedure is that the
underlying errors are all uncorrelated with homogeneous
variances. You
can test this assumption in PROC GLM by using the HOVTEST option in
the MEANS statement, requesting a homogeneity of variance test.
This section discusses the computational details behind these tests.
Note that the GLM procedure allows homogeneity of variance
testing for simple one-way models only. Homogeneity of variance testing
for more complex models is a subject of current research.

Bartlett (1937) proposes a test for equal variances that is a
modification of the normal-theory likelihood ratio test (the
HOVTEST=BARTLETT option). While Bartlett's test has accurate Type I
error rates and optimal power when the underlying distribution of the
data is normal, it can be very inaccurate if that distribution is even
slightly nonnormal (Box 1953). Therefore, Bartlett's test is not
recommended for routine use.

An approach that leads to tests that are much more robust to the
underlying distribution is to transform the original values of the
dependent variable to derive a dispersion variable and then to
perform analysis of variance on this variable. The significance level
for the test of homogeneity of variance is the p-value for the
ANOVA F-test on the dispersion variable. All of the homogeneity
of variance tests available in PROC GLM except Bartlett's use this
approach.

Levene's test (Levene 1960) is widely considered to be the standard
homogeneity of variance test (the HOVTEST=LEVENE option). Levene's
test is of the dispersion-variable-ANOVA form discussed previously, where
the dispersion variable is either

O'Brien (1979) proposes a test (HOVTEST=OBRIEN) that is basically a
modification of Levene's z2ij, using the dispersion variable

where ni is the size of the ith group and is its
sample variance. You can use the W= option in
parentheses to tune O'Brien's zWij dispersion variable to match
the suspected kurtosis of the underlying distribution. The choice of
the value of the W= option is rarely critical. By default, W=0.5, as
suggested by O'Brien (1979, 1981).

Finally, Brown and Forsythe (1974) suggest using the absolute
deviations from the group medians:

where mi is the median of the ith group. You can use the
HOVTEST=BF option to specify this test.

Simulation results (Conover, Johnson, and Johnson 1981; Olejnik and
Algina 1987) show that, while all of these ANOVA-based tests are
reasonably robust to the underlying distribution, the Brown-Forsythe
test seems best at providing power to detect variance differences
while protecting the Type I error probability. However, since the
within-group medians are required for the Brown-Forsythe test, it can
be resource intensive if there are very many groups or if some groups
are very large.

If one of these tests rejects the assumption of homogeneity of
variance, you should use Welch's ANOVA instead of the usual ANOVA to
test for differences between group means. However, this conclusion
holds only if you use one of the robust homogeneity of variance tests
(that is, not for HOVTEST=BARTLETT); even then, any homogeneity of
variance test has too little power to be relied upon always to
detect when Welch's ANOVA is appropriate. Unless the group variances
are extremely different or the number of groups is large, the usual
ANOVA test is relatively robust when the groups are all about the same
size. As Box (1953) notes, "To make the preliminary test on
variances is rather like putting to sea in a rowing boat to find out
whether conditions are sufficiently calm for an ocean liner to leave
port!"

Example 30.10 illustrates the use of the HOVTEST
and WELCH options in the MEANS statement in testing for equal group
variances and adjusting for unequal group variances
in a one-way ANOVA.